# Finite-difference modeling of Biot's poroelastic equations across all frequencies

## Abstract

An explicit time-stepping finite-difference scheme is presented for solving Biot's equations of poroelasticity across the entire band of frequencies. In the general case for which viscous boundary layers in the pores must be accounted for, the time-domain version of Darcy's law contains a convolution integral. It is shown how to efficiently and directly perform the convolution so that the Darcy velocity can be properly updated at each time step. At frequencies that are low enough compared to the onset of viscous boundary layers, no memory terms are required. At higher frequencies, the number of memory terms required is the same as the number of time points it takes to sample accurately the wavelet being used. In practice, we never use more than 20 memory terms and often considerably fewer. Allowing for the convolution makes the scheme even more stable (even larger time steps might be used) than it is when the convolution is entirely neglected. The accuracy of the scheme is confirmed by comparing numerical examples to exact analytic results.

- Authors:

- Publication Date:

- Research Org.:
- Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States)

- Sponsoring Org.:
- Earth Sciences Division

- OSTI Identifier:
- 980772

- Report Number(s):
- LBNL-2808E

Journal ID: ISSN 0016-8033; GPYSA7; TRN: US201012%%1334

- DOE Contract Number:
- DE-AC02-05CH11231

- Resource Type:
- Journal Article

- Journal Name:
- Geophysics

- Additional Journal Information:
- Journal Volume: 75; Journal Issue: N33; Related Information: Journal Publication Date: 2009; Journal ID: ISSN 0016-8033

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 54; 58; ACCURACY; BOUNDARY LAYERS; EQUATIONS; LAWS; SIMULATION; USES; VELOCITY

### Citation Formats

```
Masson, Y J, and Pride, S R.
```*Finite-difference modeling of Biot's poroelastic equations across all frequencies*. United States: N. p., 2009.
Web.

```
Masson, Y J, & Pride, S R.
```*Finite-difference modeling of Biot's poroelastic equations across all frequencies*. United States.

```
Masson, Y J, and Pride, S R. Thu .
"Finite-difference modeling of Biot's poroelastic equations across all frequencies". United States. https://www.osti.gov/servlets/purl/980772.
```

```
@article{osti_980772,
```

title = {Finite-difference modeling of Biot's poroelastic equations across all frequencies},

author = {Masson, Y J and Pride, S R},

abstractNote = {An explicit time-stepping finite-difference scheme is presented for solving Biot's equations of poroelasticity across the entire band of frequencies. In the general case for which viscous boundary layers in the pores must be accounted for, the time-domain version of Darcy's law contains a convolution integral. It is shown how to efficiently and directly perform the convolution so that the Darcy velocity can be properly updated at each time step. At frequencies that are low enough compared to the onset of viscous boundary layers, no memory terms are required. At higher frequencies, the number of memory terms required is the same as the number of time points it takes to sample accurately the wavelet being used. In practice, we never use more than 20 memory terms and often considerably fewer. Allowing for the convolution makes the scheme even more stable (even larger time steps might be used) than it is when the convolution is entirely neglected. The accuracy of the scheme is confirmed by comparing numerical examples to exact analytic results.},

doi = {},

journal = {Geophysics},

issn = {0016-8033},

number = N33,

volume = 75,

place = {United States},

year = {2009},

month = {10}

}